Note on beta elements in homotopy, and an application to the prime three case
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Abstract:
Let $S^0_{(p)}$ denote the sphere spectrum localized at an odd prime $p$. Then we have the first beta element $\beta _1\in \pi _{2p^2-2p-2}(S^0_{(p)})$, whose cofiber we denote by $W$. We also consider a generalized Smith-Toda spectrum $V_r$ characterized by $BP_*(V_r)=BP_*/(p,v_1^r)$. In this note, we show that an element of $\pi _*(V_r\wedge W)$ gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of $\beta _{9t+3}$ at the prime three to complete a conjecture of Ravenel’s: $\beta _{s}\in \pi _{16s-6}(S^0_{(3)})$ exists if and only if $s$ is not congruent to $4$, $7$ or $8$ mod $9$.References
- Mark Behrens and Satya Pemmaraju, On the existence of the self map $v^9_2$ on the Smith-Toda complex $V(1)$ at the prime 3, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 9–49. MR 2066495, DOI 10.1090/conm/346/06284
- Shichirô Oka, The stable homotopy groups of spheres. II, Hiroshima Math. J. 2 (1972), 99–161. MR 322865
- Shichirô Oka, Ring spectra with few cells, Japan. J. Math. (N.S.) 5 (1979), no. 1, 81–100. MR 614695, DOI 10.4099/math1924.5.81
- Shichirô Oka, A new family in the stable homotopy groups of spheres, Hiroshima Math. J. 5 (1975), 87–114. MR 380791
- Shichirô Oka, A new family in the stable homotopy groups of spheres. II, Hiroshima Math. J. 6 (1976), no. 2, 331–342. MR 418096
- Shichirô Oka, Realizing some cyclic $\textrm {BP}_\ast$-modules and applications to stable homotopy of spheres, Hiroshima Math. J. 7 (1977), no. 2, 427–447. MR 474290
- Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
- Katsumi Shimomura, The homotopy groups of the $L_2$-localized Toda-Smith spectrum $V(1)$ at the prime $3$, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1821–1850. MR 1370651, DOI 10.1090/S0002-9947-97-01710-8
- Larry Smith, On realizing complex bordism modules. IV. Applications to the stable homotopy groups of spheres, Amer. J. Math. 99 (1977), no. 2, 418–436. MR 433450, DOI 10.2307/2373828
- Hirosi Toda, $p$-primary components of homotopy groups. IV. Compositions and toric constructions, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1959), 297–332. MR 111041, DOI 10.1215/kjm/1250776579
- Hirosi Toda, Algebra of stable homotopy of $Z_{p}$-spaces and applications, J. Math. Kyoto Univ. 11 (1971), 197–251. MR 293631, DOI 10.1215/kjm/1250523647
- Hirosi Toda, On spectra realizing exterior parts of the Steenrod algebra, Topology 10 (1971), 53–65. MR 271933, DOI 10.1016/0040-9383(71)90017-6
Additional Information
- Katsumi Shimomura
- Affiliation: Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan
- Email: katsumi@kochi-u.ac.jp
- Received by editor(s): April 19, 2009
- Published electronically: December 8, 2009
- Communicated by: Brooke Shipley
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1495-1499
- MSC (2010): Primary 55Q45
- DOI: https://doi.org/10.1090/S0002-9939-09-10190-9
- MathSciNet review: 2578544